Article: Integrators on homogeneous spaces: Isotropy choice and connections
Authors: Hans Munthe-Kaas, Olivier Verdier
Journal: To appear in J. Found. Comput. Math.
Abstract: We consider numerical integrators of ODEs on homogeneous spaces (spheres, affine spaces, hyperbolic spaces). Homogeneous spaces are equipped a built-in symmetry. A numerical integrator respects this symmetry if it is equivariant. One obtain homogeneous space integrators by combining a Lie group integrator with an isotropy choice. We show that equivariant isotropy choices combined with equivariant Lie group integrators produces an equivariant homogeneous space integrator. Moreover, we show that the RKMK, Crouch–Grossman or commutator-free methods are equivariant. In order to show this, we give a novel description of Lie group integrators in terms of stage trees and motion maps, which unifies the known Lie group integrators. We then proceed to study the equivariant isotropy maps of order zero, which we call connections, and show that they can be identified with reductive structures and invariant principal connections. We give concrete formulas for connections in standard homogeneous spaces of interest, such as Stiefel, Grassmannian, isospectral, and polar decomposition manifolds. Finally, we show that the space of matrices of fixed rank possesses no connection.
Hindsight notes: None yet.
9 Sep 2014